Download A G2-QCD neutron star

arXiv:1609.06979v1 [astro-ph.HE] 21 Sep 2016
A G2-QCD neutron star
Ouraman Hajizadeh∗†
E-mail: [email protected]
Axel Maas
E-mail: [email protected]
Institute of Physics, NAWI Graz, University of Graz, Universitätsplatz 5, 8010 Graz, Austria
The determination of the properties of neutron stars from the underlying theory, QCD, is still an
unsolved problem. This is mainly due to the difficulty to obtain reliable results for the equation of
state for cold, dense QCD. As an alternative route to obtain qualitative insights, we determine the
structure of a neutron star for a modified version of QCD: By replacing the gauge group SU(3)
with the exceptional Lie group G2 , it is possible to perform lattice simulations at finite density,
while still retaining neutrons. Here, results of these lattice simulations are used to determine the
mass-radius relation of a neutron star for this theory. The results show that phase changes express
themselves in this relation. Also, the radius of the most massive neutron stars is found to vary
very little, which would make radius determinations much simpler if this would also be true in
QCD.
34th annual International Symposium on Lattice Field Theory
24-30 July 2016
University of Southampton, UK
∗ Speaker.
† Supported
by the FWF DK W1203-N16
c Copyright owned by the author(s) under the terms of the Creative Commons
Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0).
http://pos.sissa.it/
Ouraman Hajizadeh
A G2 -QCD neutron star
1. Introduction
The properties of neutron stars [1–3] have been the subject of astronomical studies and observations since decades. Nevertheless, a satisfying explanation of their properties is still out of
reach. The main reason for this is that it was not yet possible to derive the equation of state governing neutron stars from the fundamental theory, i. e. QCD, at finite density and small or zero
temperature [4]. The reason is that lattice gauge theory, the mainstay of non-perturbative QCD calculations, suffers from the sign problem [5, 6]. Alternative methods are either still not sufficiently
far developed or make heavy use of modeling, which reduces their predictivity [7–10].
Turning this around, data of neutron stars could provide us with a laboratory for strong interactions in this cold and dense regime of the phase diagram [1, 2, 11]. Hence, they make it
possible to observe large-scale effects of non-Abelian gauge theories. Also the recent discovery of
gravitational waves [12] will provide data on the bulk properties, giving further constraints [13].
Pending a solution for QCD, it would be highly interesting to know whether any generic signatures of a realistic equation of state of a non-Abelian gauge theory can manifest in the features
of neutron stars. To make progress, the avenue exploited here will be to study a QCD-like theory
without sign problem. To have any chance at qualitative similarity requires this theory to have
fermionic baryons, especially neutrons, and have rather similar features as QCD itself. A theory
with these feature is G2 -QCD, i. e. QCD where the gauge group SU(3) is replaced by the exceptional Lie group G2 [14, 15], see [16] for a review of the properties of this theory. A rough outline
of the full phase diagram [15], detailed spectroscopical data and properties of the structure at zero
temperature and finite baryon density [17] are available for this theory from lattice simulations.
Based on these results, we determine the mass-radius relation of a G2 -QCD neutron star. Because the data are for one flavor and comparatively heavy pions, it will probably not be a semiquantitatively good guideline, but some qualitative features can be expected, and seem to be, rather
robust. This will be discussed in more detail elsewhere [18]. There, we will also consider the
impact of varying parameters like the pion mass. In total, the features obtained are not too different
from the expectations from models [1–3], which is by itself promising. Most interesting is that
we find evidence for different phases inside this model neutron star to surface in astronomically
observable quantities.
2. G2 -QCD
We will briefly rehearse here the relevant features of G2 -QCD for our investigation [17]. G2
is a subgroup of SO(7), and has 7 quark colors and 14 gluons [14], and all representations are real.
Therefore it does not suffer from a sign problem [15, 19].
Furthermore, the theory shares many features with QCD. Among those similarities are the
coincidence of chiral and deconfinement transition in the quenched case [20]. The spectrum is
rich [17], and contains fermionic baryons [14, 17], in particular three-quark states, to be called
neutrons in the following. At sufficiently light quark masses it is expected that these are the lightest
fermionic baryons [15, 17]. At the same time, it is possible to construct color-neutral states with
any number of quarks and varying numbers of gluons [14, 17], as well as mesons and glueballs.
1
Ouraman Hajizadeh
A G2 -QCD neutron star
Due to the reality of the representations there is an enlarged chiral symmetry, the Pauli-Gürsey
symmetry [17, 19]. Thus, non-trivial symmetry breaking is possible for the one-flavor case considered here. The Goldstone bosons in this case are diquarks [17].
In the following, we use as input the lattice data for the light ensemble of [17], i. e. a Goldstone
mass of 247 MeV with a neutron of 938 MeV mass fixing the scale. For details of the simulations
we refer also to this work.
3. Mass-Radius relation of neutron stars
The most pertinent feature of neutron stars is their mass-radius relation [1]. To obtain it, we
follow the approximation of Tolman, Oppenheimer and Volkoff: We assume a static and spherically symmetric metric, and an energy-momentum tensor of an ideal fluid, which are realistic and
simplifying assumptions. This yields the Tolman-Oppenheimer-Volkoff (TOV) equation [1]
[p(r) + ε(r)] [M(r) + 4πr3 p(r)]
d p(r)
=−
,
dr
r [r − 2M(r)]
where M(r) is the mass-energy enclosed within the radius r:
Z r
M(r) = 4π
ε(r0 ) r02 dr0 ,
(3.1)
0
and p and ε are the pressure and energy density, respectively. Thus, knowledge of both quantities
are required to solve the TOV equation.
For this purpose, we will employ two different versions of the pressure and energy density.
As a baseline, we employ the well-known [1] case of non-interacting neutrons. This helps us
understanding the impact of interactions.
nb fm3
3.5
3.0
2.5
2.0
1.5
1.0
0.5
2
4
6
8
μ fm
Figure 1: The employed baryon density as a function of baryo-chemical potential, interpolated from the
lattice results of [17]. The red band shows the mass of the lightest particle in the spectrum, the diquark
Goldstone, while the green band is the mass of the neutron, both values also from [17], both including
statistical errors only.
For the full interacting theory, we neglect contributions from the kinetic energy. As the rest
mass is much larger than the average thermal energy, this should be a decent approximation. There2
Ouraman Hajizadeh
A G2 -QCD neutron star
fore, the energy density is entirely given by the density of neutrons as a function of the baryochemical potential µ, n(µ) and their rest mass mn , ε = mn n(µ), and the pressure is then [3]
p = nµ − ε = (µ − mn )n(µ).
Thus, if the baryo-chemical potential drops to the mass of the neutron, the pressure drops to zero,
and the surface of the neutron star is reached. The baryon density is taken from the lattice simulations of G2 -QCD of [17], and depicted in figure 1. Note that the lattice only yields discrete points,
which we have interpolated.
The actual baryon density already starts to differ from zero at a baryo-chemical potential below
the neutron mass, as the Goldstones are diquarks and also carry baryon number [17]. Here, we
assume that the neutron star consists entirely of neutrons, i. e. the diquarks, because of their bosonic
nature, do not contribute significantly in the chemical composition. Relaxing this assumption will
be studied elsewhere [18], but so far does not seem to have strong qualitative impacts. A further
problem is that the mesh of points in baryo-chemical potential on the lattice does not include the
point µ = mn . Around this chemical potential, the pressure must be the one of ordinary matter to
provide a stable neutron star [1], which we enforce in the fit. This yields a systematic, but slight,
quantitative dependence on where in the chemical potential this fit is enforced, but this has also
little qualitative impact. This will also be studied elsewhere in more detail [18].
Finally, rewriting the TOV equation using this pressure only and in terms of the baryo-chemical
potential yields
dµ(r)
[µ(r)] [M(r) + 4πr3 p(r)]
=−
.
(3.2)
dr
r [r − 2M(r)]
To obtain the mass-radius relation we solve the coupled equations (3.1) and (3.2) for various values
of the chemical potential at the core of the neutron star, starting from slightly above mn . The initial
conditions are obtained from the requirement of zero mass at the center of the neutron stars and the
corresponding pressure value. The solution itself is possible using standard algorithms for ordinary
differential equations.
4. Results
As figure 1 shows, the neutron star will sample only the region starting from the green band,
due to our assumption of the chemical composition. The most important feature in the baryon
density is a step-like structure around and above this point, which could be associated with a phase
change [17]. This is interpreted as the onset of a phase where neutrons or heavier fermionic hadrons
dominate [17], and which is thus precisely the relevant phase here. Of course, at very high chemical
potentials lattice artifacts will play a role [15, 17], but it turns out that this is at a value of chemical
potential where the neutron star is no longer stable.
The results for the mass-radius-relation are shown in figure 2. What is immediately seen from
the plot is that the interaction increases the maximum mass of the neutron star and also that it shows
less stability in response to perturbations in the radius, comparing to the free case. Turned around,
given an (almost) maximum mass neutron star, its radius is essentially fixed. The end-point of the
mass-radius curve is at a chemical potential where lattice artifacts start to become an issue, but are
not yet dominating.
3
Ouraman Hajizadeh
A G2 -QCD neutron star
M
M⊙
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
R
5
10
15
20
25
30 km
Figure 2: Mass-radius relation for the G2 neutron star (blue) in comparison to the result for the free Fermi
gas (orange).
At large radius the free neutron gas allows larger masses, than the interacting case, which
leads to a steeper rise. Conversely, at a smaller radius, about 11 km, the rise for the interacting
case becomes much quicker. This can be traced back to the baryon density shown in figure 1.
The corresponding central chemical potential coincides with a region slightly above the masses of
ground state neutrons and ∆s. Starting from this point, the equation of state appears to enter a new
region dominated by fermionic hadrons [17]. Thus, this rapid change of properties impresses itself
on the mass-radius relation. This could imply that changes of slope of the mass-radius relation of
neutron stars could also be related to different phases.
It is also interesting to look inside the G2 neutron star to see, whether in their profile the
different phases are distinguished. This is studied in figure 3. In fact, the change is seen also in the
pressure profile , though the effect is much less pronounced than in the density itself. However, the
mass distribution is not visibly affected.
The different profiles also explain why the radius distribution around the maximum mass is
much narrower than in the free case. In figure 4 the profiles of µ(r) curves do not diverge considerably after the crossing point, in comparison to the free gas, while they are quickly changing in the
interacting gas. Also, a much higher central chemical potential is supported than for the free gas,
leading to the larger mass. It is also visible how the slope is negative throughout, thus supporting
the stability of the star.
4
Ouraman Hajizadeh
A G2 -QCD neutron star
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
2
4
6
8
r
km
10
1.2
1.0
0.8
M(r)/M⊙
0.6
p(r)/pc
0.4
0.2
0.0
0
r
2
4
6
8
10km
Figure 3: Accumulated mass (blue) and pressure (orange) as a function of distance from the center of the
neutron star for two different central chemical potential, µc = 7.6 fm−1 (left) and µc = 6 fm−1 (right). The
green line indicates the last step before the neutron threshold in the baryon density in figure 1.
Figure 4: The chemical potential as a function of distance from the center for different values of the central
chemical potential for G2 -QCD (left) and for the free neutron gas (right)
5. Conclusions
We have provided, for the first time, a mass-radius relation for a neutron star based on the
equation of state of a QCD-like theory with fermionic baryons, using lattice data as an input.
The results show a number of expected features. In particular, the curves become steeper
and the maximum mass increases, without exceeding the maximum value possible from general
relativity. Of course, given the fact that the theory is different from QCD and that only one flavor
with a rather heavy Goldstone is included, the increase to roughly 1.2 solar masses instead of the
more than two possible in QCD is not entirely unexpected.
However, even this could already hint that either the masses of the quarks or the number of
degrees of freedom play a significant role for the maximum neutron star mass. Less speculative
and more visible in the data is that the phase change seen in the baryon density impresses itself into
the mass-radius relation of the neutron star. It also survives in the pressure gradient. This implies
that already observing the mass-radius relation in sufficient detail could resolve the old question of
whether a neutron star is a, more or less, monolithic object, or whether it contains a multitude of
different phases.
These kind of qualitative insights show how important the study of QCD-like theories is for the
understanding of possible features of the real case. Of course, the results here could be improved in
5
Ouraman Hajizadeh
A G2 -QCD neutron star
various ways, some of which we will investigate in [18], while others will have to await improved
results for the input equation of state. Still, this gives eventually the hope that we could learn how
to interpret qualitative features of neutron stars from such model systems.
References
[1] N. K. Glendenning, Compact stars: Nuclear physics, particle physics, and general relativity
(, 1997).
[2] A. W. Steiner, J. M. Lattimer, and E. F. Brown, Astrophys. J. 722, 33 (2010), 1005.0811.
[3] J. I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applications (Cambridge University Press, Cambridge, 2006).
[4] B. Friman et al., Lect.Notes Phys. 814, 1 (2011).
[5] C. Gattringer and C. B. Lang, Quantum chromodynamics on the lattice (Lect. Notes Phys.,
2010).
[6] P. de Forcrand, PoS LAT2009, 010 (2009), 1005.0539.
[7] S. Leupold et al., Lect.Notes Phys. 814, 39 (2011).
[8] M. Buballa, Phys.Rept. 407, 205 (2005), hep-ph/0402234.
[9] J. M. Pawlowski, AIP Conf.Proc. 1343, 75 (2010), 1012.5075.
[10] J. Braun, J.Phys.G G39, 033001 (2012), 1108.4449.
[11] J. M. Lattimer and M. Prakash, Phys. Rept. 442, 109 (2007), astro-ph/0612440.
[12] Virgo, LIGO Scientific, B. P. Abbott et al., Phys. Rev. Lett. 116, 061102 (2016), 1602.03837.
[13] W. Del Pozzo, T. G. F. Li, M. Agathos, C. Van Den Broeck, and S. Vitale, Phys. Rev. Lett.
111, 071101 (2013), 1307.8338.
[14] K. Holland, P. Minkowski, M. Pepe, and U. J. Wiese, Nucl. Phys. B668, 207 (2003), heplat/0302023.
[15] A. Maas, L. von Smekal, B. Wellegehausen, and A. Wipf, Phys.Rev. D86, 111901 (2012),
1203.5653.
[16] A. Maas and B. H. Wellegehausen, PoS LATTICE2012, 080 (2012), 1210.7950.
[17] B. H. Wellegehausen, A. Maas, A. Wipf, and L. von Smekal, Phys.Rev. D89, 056007 (2014),
1312.5579.
[18] O. Hajizadeh and A. Maas, unpublished.
[19] J. Kogut, M. A. Stephanov, D. Toublan, J. Verbaarschot, and A. Zhitnitsky, Nucl.Phys. B582,
477 (2000), hep-ph/0001171.
[20] J. Danzer, C. Gattringer, and A. Maas, JHEP 01, 024 (2009), 0810.3973.
6